Math Guide

Incenter of a Triangle

Incenterof a triangle is defined as the point of intersection of the internal bisectors of a triangle. In the triangle ABC shown below, when we draw bisectors for all the three angles, the three bisectors meet eachother at a point. This point is called as the incenter of the triangle. In thefigure shown below, the three lines are the bisectors of the triangle and they meet at a point r, which is the incenter of the triangle.

Incenter


Steps to Find Incenter


 

Given below are the steps to find the incenter of a triangle:

1. Draw a triangle of any dimension

solveincenter

2. Using a compass draw two arcs on two successive sides of the triangle from a vertex. There are no defined width for the arcs.

 solveincenter

3.Without changing the compass reading, draw another two intersecting arcs inside the triangle from the arcs previously drawn. Now, we get an intersecting point inside the triangle.


solveincenter



4.Join the vertex and the point of intersection between the arcs. The line joining the vertex and the point of intersection is called as the angular bisector of the corresponding vertex.

 solveincenter

5. Again using the compass, draw two arcs of any length on the sides of the triangle from next vertex of the triangle.

solveincenter

6.Without changing the compass reading, draw another two intersecting arcs inside the triangle from the arcs previously drawn. Now, we get an intersecting point inside the triangle.

solveincenter

7. Join the corresponding vertex and the point of intersection between the arcs. The line is the angular bisector of the vertex.

solveincenter

8.Now the point of intersection between the two angular bisectors is called as the incenter of the triangle. Thus the incenter of the triangle is solved using angular bisector.

solveincenter

 


Properties of Incenter


 

Listed below are the Properties of Incenter:

 

  • Incenter is equidistant from all the sides of the triangle. 
  • Thebiggest circle that can be drawn inside a triangle is the incircle. Theincircle will touch all sides of the triangle. The radius of the circleis the length of the perpendicular line drawn from the Incenter to any side. 

 

PropertiesOfIncenter

 

  • For an equilateral triangle, incentre, circumcentre and centroid will all be the same. This is because the internal bisector is also a perpendicular bisector for the opposite side.

 

 


Incenter Pictures


 

Givenbelow are the pictures of incentres of acute, right angled and obtuse triangle. Unlike circumcentre, incenter always lies inside the triangle.

 

See the incentres 'D'of all the three types of triangles.IncenterOfacuteAngledTriangleIncenterOfRightAnglesTriangle

IncenterOfObtuseAngledTriangle